We consider a discrete almost periodic ratio-dependent Leslie system with time delays and feedback controls. Sufficient conditions are obtained for the permanence and global attractivity of the system. Furthermore, by using an almost periodic functional Hull theory, we show that the almost periodic system has a unique globally attractive positive almost periodic solution.

Among the relationships between the species living in the same outer environment, the predator-prey theory plays an important and fundamental role. The predator-prey models have been extensively studied by many scholars [

However, in the study of the dynamic behaviors of predator-prey system, many scholars argued that the ratio-dependent predator-prey systems are more realistic [

Though much progress has been seen in the asymptotic convergence of solutions of population systems, such systems are not well studied in the sense that most results are continuous-time cases related. Already, many scholars have paid attention to the nonautonomous discrete population models, since the discrete time models governed by difference equation are more appropriate than the continuous ones when the populations have a short life expectancy, nonoverlapping generations in the real world (see [

Since time delays occur so often in nature, a number of ecological systems can be described as systems with time delays (see [

So it is very interesting to study dynamics of the following discrete ratio-dependent Leslie system with time delays and feedback controls:

In this paper, we are concerned with the effects of the almost periodicity of ecological and environmental parameters and time delays on the global dynamics of the discrete ratio-dependent Leslie systems with feedback controls. To do so, for system (

Let

One can easily show that the solutions of system (

The principle aim of this paper is to study the dynamic behaviors of system (

The organization of this paper is as follows. In the next section, we introduce some definitions and several useful lemmas. In Section

In this section, we will introduce some basic definitions and several useful lemmas.

System (

Suppose that

A sequence

The hull of

Assume that

Assume that

Assume that

Assume that

In this section, we establish a permanent result for system (

Assume that

Let

For any

For any

By the conditions

Substituting (

This implies for any integer

Consequently, combining (

Firstly, we prove two lemmas which will be useful to our main result.

For any two positive solutions

It follows from the first equation of system (

For any two positive solutions

The proofs of Lemma

Now we are in the position of stating the main result on the global attractivity of system (

In addition to

For two arbitrary nontrivial solutions

In this section, we consider the almost periodic property of system (

We assume that

According to the almost periodic theory, we know that if system (

The following Lemma is Lemma 4.1 in [

If each hull equation of system (

Assume that

We divided the proof into two steps.

By the almost periodicity of the parameters of system (

Suppose that

For any

Let

Thus we can easily see that

Suppose that the Hull equation (

Similar to the discussion of (

Let

It follows from (

By the above discussion and Theorem

In this paper, a discrete almost periodic ratio-dependent Leslie system with time delays and feedback controls is considered. By applying the difference inequality, some sufficient conditions are established, which are independent of feedback control variables, to ensure the permanence of system (

We would like to mention here the question of whether the feedback control variables have the influence on the stability property of the system or not. It is, in fact, a very challenging problem, and we leave it for our future work.

The authors are grateful to the editor, Wolfgang Ruess, and referees for a number of helpful suggestions that have greatly improved our original submission. This work is supported by the National Natural Science Foundation of China (nos. 71171035, 71173029), China Postdoctoral Science Foundation (no. 2012M511598), and the Teaching and Research Foundation of Dongbei University of Finance and Economics (nos. YY11001, GW11003).